Electromagnetically induced grating with second field quantization in spherical semiconductor quantum dots

Abstract

A new approach for diffracting the weak probe beam into higher-order directions is proposed via electromagnetically induced grating in second field quantization formalism, offering a new way for implementations of quantum information with semiconductor quantum dots. The formalism of second field quantization allows describing atoms and photons as a many-body system. An induced diffraction grating is formed based on the electromagnetic induced transparency when a standing-wave coupling field is applied to a spherical quantum dot as a three-level system. Due to phase modulation, the zeroth-order light intensity becomes weak, and the first-order diffraction is improved affectedly. On the contrary, the probe beam is barely diffracted via absorption modulation. The simulation results verify that photon numbers of probe and control fields, as well as other parameters in the QD, can lead to the diffraction efficiency of phase grating to be improved. Phase diffraction grating accompanied with a high transmissivity is demonstrated, and the first-order diffraction efficiency reaches $30\%$. Also, the impact of QD dimensions on its optical response is investigated. This model may find potential applications in designing the semiconductor quantum dot-based photonic devices in optical communications and quantum information networks.

Appendix: The derivation of the induced dipole moments of the QD

By applying the spherical potential for the QD with radius $R_{s}$, its unperturbed Hamiltonian is written as

$$\begin{aligned} H_{0} = - \frac{-\hbar ^{2}}{2m_{e}^{*}}\triangledown ^{2} - \frac{1}{2} m_{e}^{*}\omega _{0}^{2}r^{2} \end{aligned}$$

(15)

here $m_{e}^{*}$ is the effective mass of electron, r is the spherical radial coordinate, $\omega _{0} = \hbar /(2m_{e}^{*}R_{s}^{2})$ is the electron oscillation frequency in the direction of QD radius and $k = \sqrt{m_{e}^{*}\omega _{0}/\hbar ^{2}}$ is the effective wave number.

By considering the separation of variables method, eigenfunction of the Hamiltonian $H_{0}$ can be calculated as

$$\begin{aligned} \psi _{n_{1},n_{2},n_{3}}(x_{1}, x_{2}, x_{3}) & = \sqrt{\frac{k^{3}}{2^{(n_{1}+n_{2}+n_{3})}n_{1}!n_{2}!n_{3}!\sqrt{\pi ^{3}}}}\nonumber \\&\quad\times \,H_{n_{1}}(x_{1})H_{n_{2}}(x_{2})H_{n_{3}}(x_{3}), \end{aligned}$$

(16)

where $n_{i} (i =1,2,3)$ are quantum numbers and $H_{n_{i}}(x_{i})$ the $n_{i}$th Hermitian polynomial. The eigenvalues of $H_{0}$ is also obtained by

$$\begin{aligned} E^{0}_{n_{1},n_{2},n_{3}}=\left( n_{1}+n_{2}+n_{3}+\frac{3}{2}\right) \hbar \omega _{0} \end{aligned}$$

(17)

By considering the forbidden and allowed transitions,The induced dipole moments for transitions $|{1}\rangle \leftrightarrow |{2}\rangle (\mu _{12})$ and $|{1}\rangle \leftrightarrow |{3}\rangle (\mu _{13})$ are calculated as

$$\begin{aligned} \mu ^{z}_{12}(R_{s}) & = -\,4eR_{s}^{5}k^{4}e^{-k^{2}R_{s}^{2}}/5\sqrt{3\pi }\nonumber \\ \mu ^{z}_{13}(R_{s}) & = (e/35\sqrt{\pi })\left(\frac{105erf(R_{s}k)}{k}\right.\nonumber \\&\quad- \left.\,2R_{s}e^{-R_{s}^{2}k^{2}}\times \frac{(105 +70R_{s}^{2}k^{2} +14R_{s}^{4}k^{4} +20R_{s}^{6}k^{6} )}{\sqrt{\pi }}\right) \end{aligned}$$

(18)

where erf(x) is the error function.